Monotone drawings of graphs with few directions

We show that a planar graph admits a monotone drawing with only one direction of monotonicity if and only if it has a Hamiltonian path between two vertices lying in the same face. Also, we prove that maximal planar graphs admit monotone drawings with two (orthogonal) directions, while triconnected planar graphs with three directions. The latter two results are obtained by applying the famous drawing algorithm based on Schnyder realizers.